Leture 13: Weak onvergene
A sequene
f
P
n
g
of probability measures on (
R
k
;
B
k
) is
tight
if for every
>
0, there is a
ompat set
C
R
k
suh that inf
n
P
n
(
C
)
>
1
.
If
f
X
n
g
is a sequene of random
k
-vetors, then the tightness of
f
P
X
n
g
is the same as the
boundedness of
fk
X
n
kg
in probability (
k
X
n
k
=
O
p
(1)).
Proposition 1.17.
Let
f
P
n
g
be a sequene of probability measures on (
R
k
;
B
k
).
(i) Tightness of
f
P
n
g
is a neessary and suÆient ondition that for every subsequene
f
P
n
i
g
there exists a further subsequene
f
P
n
j
g f
P
n
i
g
and a probability measure
P
on (
R
k
;
B
k
)
suh that
P
n
j
!
w
P
as
j
! 1
.
(ii) If
f
P
n
g
is tight and if eah subsequene that onverges weakly at all onverges to the
same probability measure
P
, then
P
n
!
w
P
.
The proof an be found in Billingsley (1986, pp. 392-395).
The following result gives some useful suÆient and neessary onditions for onvergene in
distribution.
Theorem 1.9.
Let
X; X
1
; X
2
;:::
be random
k
-vetors.
(i)
X
n
!
d
X
is equivalent to any one of the following onditions:
(a)
E
[
h
(
X
n
)℄
!
E
[
h
(
X
)℄ for every bounded ontinuous funtion
h
;
(b) lim sup
n
P
X
n
(
C
)
P
X
(
C
) for any losed set
C
R
k
;
() lim inf
n
P
X
n
(
O
)
P
X
(
O
) for any open set
O
R
k
.
(ii) (Levy-Cramer ontinuity theorem). Let
X
;
X
1
;
X
2
; :::
be the h.f.'s of
X; X
1
; X
2
; :::
,
respetively.
X
n
!
d
X
if and only if lim
n
!1
X
n
(
t
) =
X
(
t
) for all
t
2 R
k
.
(iii) (Cramer-Wold devie).
X
n
!
d
X
if and only if
X
n
!
d
X
for every
2 R
k
.
Proof.
(i) First, we show
X
n
!
d
X
implies (a). By Theorem 1.8(iv) (Skorohod's theorem),
there exists a sequene of random vetors
f
Y
n
g
and a random vetor
Y
suh that
P
Y
n
=
P
X
n
for all
n
,
P
Y
=
P
X
and
Y
n
!
a:s:
Y
. For bounded ontinuous
h
,
h
(
Y
n
)
!
a:s:
h
(
Y
) and, by the
dominated onvergene theorem,
E
[
h
(
Y
n
)℄
!
E
[
h
(
Y
)℄. Then (a) follows from
E
[
h
(
X
n
)℄ =
E
[
h
(
Y
n
)℄ for all
n
and
E
[
h
(
X
)℄ =
E
[
h
(
Y
)℄.
Next, we show (a) implies (b). Let
C
be a losed set and
f
C
(
x
) = inf
fk
x
y
k
:
y
2
C
g
.
Then
f
C
is ontinuous. For
j
= 1
;
2
; :::
, dene
'
j
(
t
) =
I
(
1
;
0℄
+ (1
jt
)
I
(0
;j
1
℄
. Then
h
j
(
x
) =
'
j
(
f
C
(
x
)) is ontinuous and b ounded,
h
j
h
j
+1
,
j
= 1
;
2
; :::
, and
h
j
(
x
)
!
I
C
(
x
) as
j
! 1
. Hene lim sup
n
P
X
n
(
C
)
lim
n
!1
E
[
h
j
(
X
n
)℄ =
E
[
h
j
(
X
)℄ for eah
j
(by (a)). By
the dominated onvergene theorem,
E
[
h
j
(
X
)℄
!
E
[
I
C
(
X
)℄ =
P
X
(
C
). This proves (b).
For any open set
O
,
O
is losed. Hene, (b) is equivalent to (). Now, we show (b)
and () imply
X
n
!
d
X
. For
x
= (
x
1
; :::; x
k
)
2 R
k
, let (
1
; x
℄ = (
1
; x
1
℄
(
1
; x
k
℄ and (
1
; x
) = (
1
; x
1
)
(
1
; x
k
). From (b) and (),
P
X
((
1
; x
))
lim inf
n
P
X
n
((
1
; x
))
lim inf
n
F
X
n
(
x
)
lim sup
n
F
X
n
(
x
) = lim sup
n
P
X
n
((
1
; x
℄)
P
X
((
1
; x
℄) =
F
X
(
x
). If
x
is a ontinuity point of
F
X
, then
P
X
((
1
; x
)) =
F
X
(
x
). This
proves
X
n
!
d
X
and ompletes the pro of of (i).
(ii) From (a) of part (i),
X
n
!
d
X
implies
X
n
(
t
)
!
X
(
t
), sine
e
p
1
t
x
= os(
t
x
) +
p
1 sin(
t
x
) and os(
t
x
) and sin(
t
x
) are bounded ontinuous funtions for any xed
t
.
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